Romberg求积分公式Word文档格式.docx
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在等距节点的情况下，通过对求积区间（a，b）的逐次分半，由梯形公式出可逐次提高求积公式精度，这就是Romberg求积的基本思路，由于梯形公式余项只有
精度，即
，但当节点加密时可组合成
其精度达到
，如果再由
与
组合成
则可使误差精度达到
，于是
依赖于x，若
在
上各阶导数存在，将
展开，可将
展成
的幂级数形式，即
，记
的计算精度，可利用外推原理逐次消去式
右端
只要将步长h逐次分半，利用
及
组合消去
，重复同一过程最后可得到递推公式
，此时
.说明用
其误差阶为
，这里
表示m次加速。
计算时用序列
表示区间分半次数，即
具体计算公式为
，就是Romberg求积方法。
2、程序代码：
M文件
1）、Romberg加速法
function[s,n]=rbg1（a,b,eps）
ifnargin<
3,eps=1e-6;
end
s=10;
s0=0;
k=2;
t（1,1）=（b-a）*（f（a）+f（b））/2;
while（abs（s-s0）>
eps）
h=（b-a）/2^（k-1）;
w=0;
if（h~=0）
fori=1:
（2^（k-1）-1）
w=w+f（a+i*h）;
end
t（k,1）=h*（f（a）/2+w+f（b）/2）;
forl=2:
k
（k-l+1）
t（i,l）=（4^（l-1）*t（i+1,l-1）-t（i,l-1））/（4^（l-1）-1）;
s=t（1,k）;
s0=（t（1,k-1））;
k=k+1;
n=k;
elses=s0;
n=-k;
2）、改进的Romberg求积函数
function[s,eer]=rbg2（a,b,eps）
m=1;
r（1,1）=0;
while（（abs（t（1,m）-r（1,m））/2）>
c=0;
m=m+1;
forj=1:
2^（m-1）
c=c+f（a+（j-0.5）*（b-a）/2^（m-1））;
r（m,1）=（b-a）*c/2^（m-1）;
forj=2:
m
fork=1:
（m-j+1）
r（k,j）=r（k+1,j-1）+（r（k+1,j-1）-r（k,j-1））/（4^（j-1）-1）;
t（1,j）=r（1,j-1）+2*（4^（j-2）-1）*（t（1,j-1）-r（1,j-1））/（4^（j-1）-1）;
err=abs（t（1,m）-r（1,m））/2;
s=t（1,m）;
3）定义f.m函数如下：
functionf=f（x）;
f=x.^3;
4）运行命令及结果
>
rbg1（0,2）
rbg2（0,2）
3、流程图
二、圆柱体问题
1、问题分析
圆柱体水平方向受到地面的摩擦阻力f*Q，该摩擦力对轴心速度起减速作用，同时又产生一个力矩，对角速度起加速作用。
综上，等到轴心速度v=w*r时，圆柱体将无摩擦运动。
2、源程序：
r=input（'
r='
）;
Q=input（'
Q='
g=input（'
g='
f=input（'
f='
v0=input（'
v0='
w0=input（'
w0='
ifv0<
r*w0;
end;
j=Q*r^2/2/g;
F=f*Q;
beta=F*r/j;
a=-F/（Q/g）;
t=（v0-w0*r）/（beta*r-a）
v=v0+a*t
3、执行命令
move
r=1
Q=100
g=9.81
f=0.1
v0=3
w0=2
t=
0.3398
v=
2.6667
三、高程
1、题中已给出4*4=16个数据，分别对应16个坐标位置上的高程，现只需采用插值的方法，向其中填补数值，便可拟合对应的曲面，考虑到找到适合曲面的二元函数比较复杂，并且插值之后的数据量够大（10000个），具有一定的代表性，因此在求解丘陵最高点及其高程的时候，可以将所有数据进行比较，取其最大值所对应的x，y值作为最高点。
具体程序如下
function[s,x0,y0]=high（N）
x=[100200300400];
y=[100200300400]'
;
z=[636697624478;
698712630478;
680674598412;
662626552334];
xx=linspace（100,400,N）;
yy=linspace（100,400,N）'
zh=interp2（x,y,z,xx,yy,'
cubic'
）
mesh（xx,yy,zh）
s=0;
fori=1:
N^2
ifzh（i）>
s
s=zh（i）;
n=mod（i,N）;
m=（i-n）/N;
x0=100+300/N*m
y0=100+300/N*n
high（100）
运行结果如下：
作出拟合曲面为
求得结果：
zh=
Columns1through7
636.0000639.8173643.5115647.0826650.5308653.8558657.0579
639.0542642.7968646.4175649.9164653.2935656.5488659.6822
642.0349645.7037649.2519652.6794655.9864659.1727662.2384
644.9421648.5381652.0146655.3717658.6094661.7276664.7265
647.7759651.2999654.7057657.9932661.1625664.2136667.1464
650.5363653.9892657.3251660.5440663.6458666.6305669.4982
653.2231656.6060659.8729663.0240666.0591668.9784671.7818
655.8365659.1502662.3491665.4332668.4027671.2574673.9974
658.3765661.6218664.7536667.7717670.6763673.4673676.1448
660.8430664.0209667.0864670.0395672.8801675.6083678.2240
663.2360666.3475669.3476672.2365675.0140677.6802680.2352
665.5556668.6015671.5372674.3627677.0781679.6832682.1781
667.8017670.7829673.6551676.4182679.0722681.6172684.0530
669.9743672.8918675.7014678.4029680.9965683.4821685.8597
672.0735674.9282677.6760680.3169682.8510685.2781687.5983
674.0992676.8920679.5790682.1602684.6355687.0051689.2688
676.0514678.7833681.4103683.9326686.3502688.6630690.8711
677.9302680.6020683.1700685.6344687.9950690.2520692.4053
679.7355682.3481684.8581687.2654689.5700691.7720693.8713
681.4674684.0217686.4745688.8256691.0751693.2230695.2692
683.1258685.6228688.0192690.3150692.5103694.6049696.5990
684.7107687.1513689.4923691.7338693.8756695.9179697.8607
686.2222688.6073690.8938693.0817695.1711697.1619699.0542
687.6602689.9907692.2236694.3589696.3967698.3369700.1796
689.0248691.3016693.4818695.5654697.5524699.4429701.2368
690.3159692.5399694.6683696.7011698.6383700.4799702.2259
691.5335693.7057695.7832697.7661699.6543701.4479703.1469
692.6777694.7989696.8264698.7603700.6004702.3469703.9997
693.7484695.8196697.7980699.6837701.4767703.1769704.7844
694.7456696.7677698.6979700.5364702.2831703.9379705.5010
695.6694697.6433699.5262701.3184703.0196704.6300706.1494
696.5197698.4463700.2829702.0295703.6862705.2530706.7297
697.2966699.1768700.9679702.6700704.2830705.8070707.2419
698.0000699.8347701.5813703.2397704.8099706.2920707.6860
698.5943700.3851702.0886703.7048705.2338706.6754708.0298
699.0484700.7973702.4598704.0359705.5255706.9287708.2455
699.3689701.0779702.7013704.2391705.6913707.0580708.3390
699.5625701.2334702.8196704.3209705.7375707.0693708.3164
699.6359701.2705702.8211704.2877705.6703706.9689708.1835
699.5958701.1956702.7122704.1456705.4958706.7628707.9465
699.4488701.0154702.4995703.9012705.2204706.4571707.6114
699.2017700.7364702.1893703.5606704.8501706.0580707.1841
698.8610700.3651701.7882703.1303704.3914705.5715706.6706
698.4335699.9082701.3025702.6165703.8503705.0038706.0770
697.9259699.3721700.7387702.0257703.2331704.3610705.4092
697.3449698.7635700.1033701.3641702.5461703.6492704.6734
696.6970698.0889699.4026700.6381701.7954702.8745703.8754
695.9890697.3549698.6432699.8540700.9873702.0430703.0212
695.2276696.5680697.8316699.0182700.1280701.1609702.1170
694.4195695.7349696.9740698.1370699.2237700.2343701.1686
693.5713694.8620696.0771697.2167698.2807699.2692700.1821
692.6897693.9559695.1472696.2636697.3051698.2718699.1635
691.7814693.0232694.1908695.2841696.3033697.2482698.1189
690.8530692.0705693.2143694.2846695.2813696.2045697.0541
689.9113691.1042692.2243693.2713694.2455695.1468695.9752
688.9630690.1311691.2270692.2507693.2021694.0813694.8883
688.0146689.1576690.2291691.2290692.1573693.0141693.7992
687.0729688.1903689.2369690.2125691.1173691.9512692.7141
686.1445687.2358688.2568689.2077690.0883690.8987691.6390
685.2362686.3006687.2954688.2208689.0766689.8629690.5797
684.3546685.3913686.3591687.2582688.0884688.8498689.5424
683.5064684.5144685.4543686.3262687.1299687.8655688.5331
682.6983683.6766684.5875685.4311686.2073686.9162687.5577
681.9369682.8843683.7651684.5793685.3269686.0079686.6223
681.2289682.1442682.9936683.7772684.4949685.1468685.7328
680.5811681.4628682.2794683.0310683.7175684.3389684.8952
680.0000680.8466681.6290682.3471683.0009683.5904684.1157
679.4545680.2651681.0123681.6959682.3161682.8729683.3661
678.9091679.6833680.3948681.0437681.6299682.1535682.6144
678.3636679.1011679.7767680.3904680.9423681.4324681.8607
677.8182678.5185679.1578679.7361680.2533680.7096681.1048
677.2727677.9355678.5382679.0806679.5629679.9850680.3469
676.7273677.3522677.9179678.4242678.8711679.2587679.5869
676.1818676.7686677.2969677.7666678.1779678.5306678.8248
675.6364676.1846676.6751677.1080677.4832677.8008678.0607
675.0909675.6002676.0527676.4483676.7872677.0692677.2944
674.5455675.0155675.4295675.7876676.0897676.3359676.5261
674.0000674.4304674.8056675.1258675.3909675.6009675.7557
673.4545673.8449674.1810674.4629674.6906674.8641674.9833
672.9091673.2591673.5557673.7990673.9890674.1255674.2087
672.3636672.6729672.9297673.1340673.2859673.3852673.4321
671.8182672.0864672.3030672.4680672.5814672.6432672.6534
671.2727671.4995671.6756671.8009671.8755671.8994671.8726
670.7273670.9123671.0474671.1327671.1682671.1539671.0897
670.1818670.3246670.4185670.4635670.4595670.4066670.3048
669.6364669.7367669.7890669.7932669.7494669.6576669.5177
669.0909669.1483669.1587669.1218669.0379668.9068668.7286
668.5455668.5597668.5277668.4494668.3250668.1543667.9374
668.0000667.9706667.8959667.7759667.6107667.4001667.1442
667.4545667.3812667.2635667.1014666.8949666.6441666.3488
666.9091666.7914666.6303666.4258666.1778665.8863665.5514
666.3636666.2013665.9965665.7491665.4592665.1268664.7519
665.8182665.6108665.3619665.0714664.7393664.3656663.9503
665.2727665.0200664.7266664.3926664.0179663.6026663.1466
664.7273664.4288664.0906663.7127663.2952662.8379662.3409
664.1818663.8372663.4539663.0318662.5710662.0714661.5331
663.6364663.2453662.8165662.3498661.8454661.3032660.7232
663.0909662.6530662.1783661.6668661.1184660.5332659.9112
662.5455662.0604661.5395660.9827660.3900659.7615659.0971
662.0000661.4674660.8999660.2975659.6602658.9881658.2810
………………………………
Zh为100*100的矩阵，此处只列出部分值，其余省略。
并解得最高点和最高程为：
x0=
166
y0=
196
ans=
718.1243
即最高点在坐标（166,196）处，且高程为718.1243m。
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